Properties

Label 1344.4.a.bj
Level $1344$
Weight $4$
Character orbit 1344.a
Self dual yes
Analytic conductor $79.299$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(79.2985670477\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 672)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{17}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -3 q^{3} + ( 5 - \beta ) q^{5} + 7 q^{7} + 9 q^{9} +O(q^{10})\) \( q -3 q^{3} + ( 5 - \beta ) q^{5} + 7 q^{7} + 9 q^{9} + ( 11 + 11 \beta ) q^{11} + ( 12 - 2 \beta ) q^{13} + ( -15 + 3 \beta ) q^{15} + ( -75 + 11 \beta ) q^{17} + ( -4 - 32 \beta ) q^{19} -21 q^{21} + ( -41 - 33 \beta ) q^{23} + ( -83 - 10 \beta ) q^{25} -27 q^{27} + ( 18 + 32 \beta ) q^{29} + ( 44 - 20 \beta ) q^{31} + ( -33 - 33 \beta ) q^{33} + ( 35 - 7 \beta ) q^{35} + ( -144 + 14 \beta ) q^{37} + ( -36 + 6 \beta ) q^{39} + ( -193 + 61 \beta ) q^{41} + ( 172 + 76 \beta ) q^{43} + ( 45 - 9 \beta ) q^{45} + ( -138 + 110 \beta ) q^{47} + 49 q^{49} + ( 225 - 33 \beta ) q^{51} + ( -80 - 18 \beta ) q^{53} + ( -132 + 44 \beta ) q^{55} + ( 12 + 96 \beta ) q^{57} + ( 538 + 46 \beta ) q^{59} + ( -78 - 148 \beta ) q^{61} + 63 q^{63} + ( 94 - 22 \beta ) q^{65} + ( 686 + 6 \beta ) q^{67} + ( 123 + 99 \beta ) q^{69} + ( -551 - 55 \beta ) q^{71} + ( -120 - 178 \beta ) q^{73} + ( 249 + 30 \beta ) q^{75} + ( 77 + 77 \beta ) q^{77} + ( -206 + 106 \beta ) q^{79} + 81 q^{81} + ( 232 - 164 \beta ) q^{83} + ( -562 + 130 \beta ) q^{85} + ( -54 - 96 \beta ) q^{87} + ( -873 - 51 \beta ) q^{89} + ( 84 - 14 \beta ) q^{91} + ( -132 + 60 \beta ) q^{93} + ( 524 - 156 \beta ) q^{95} + ( 428 + 50 \beta ) q^{97} + ( 99 + 99 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{3} + 10q^{5} + 14q^{7} + 18q^{9} + O(q^{10}) \) \( 2q - 6q^{3} + 10q^{5} + 14q^{7} + 18q^{9} + 22q^{11} + 24q^{13} - 30q^{15} - 150q^{17} - 8q^{19} - 42q^{21} - 82q^{23} - 166q^{25} - 54q^{27} + 36q^{29} + 88q^{31} - 66q^{33} + 70q^{35} - 288q^{37} - 72q^{39} - 386q^{41} + 344q^{43} + 90q^{45} - 276q^{47} + 98q^{49} + 450q^{51} - 160q^{53} - 264q^{55} + 24q^{57} + 1076q^{59} - 156q^{61} + 126q^{63} + 188q^{65} + 1372q^{67} + 246q^{69} - 1102q^{71} - 240q^{73} + 498q^{75} + 154q^{77} - 412q^{79} + 162q^{81} + 464q^{83} - 1124q^{85} - 108q^{87} - 1746q^{89} + 168q^{91} - 264q^{93} + 1048q^{95} + 856q^{97} + 198q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.56155
−1.56155
0 −3.00000 0 0.876894 0 7.00000 0 9.00000 0
1.2 0 −3.00000 0 9.12311 0 7.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.a.bj 2
4.b odd 2 1 1344.4.a.br 2
8.b even 2 1 672.4.a.j yes 2
8.d odd 2 1 672.4.a.e 2
24.f even 2 1 2016.4.a.q 2
24.h odd 2 1 2016.4.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.a.e 2 8.d odd 2 1
672.4.a.j yes 2 8.b even 2 1
1344.4.a.bj 2 1.a even 1 1 trivial
1344.4.a.br 2 4.b odd 2 1
2016.4.a.q 2 24.f even 2 1
2016.4.a.r 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1344))\):

\( T_{5}^{2} - 10 T_{5} + 8 \)
\( T_{11}^{2} - 22 T_{11} - 1936 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 3 + T )^{2} \)
$5$ \( 8 - 10 T + T^{2} \)
$7$ \( ( -7 + T )^{2} \)
$11$ \( -1936 - 22 T + T^{2} \)
$13$ \( 76 - 24 T + T^{2} \)
$17$ \( 3568 + 150 T + T^{2} \)
$19$ \( -17392 + 8 T + T^{2} \)
$23$ \( -16832 + 82 T + T^{2} \)
$29$ \( -17084 - 36 T + T^{2} \)
$31$ \( -4864 - 88 T + T^{2} \)
$37$ \( 17404 + 288 T + T^{2} \)
$41$ \( -26008 + 386 T + T^{2} \)
$43$ \( -68608 - 344 T + T^{2} \)
$47$ \( -186656 + 276 T + T^{2} \)
$53$ \( 892 + 160 T + T^{2} \)
$59$ \( 253472 - 1076 T + T^{2} \)
$61$ \( -366284 + 156 T + T^{2} \)
$67$ \( 469984 - 1372 T + T^{2} \)
$71$ \( 252176 + 1102 T + T^{2} \)
$73$ \( -524228 + 240 T + T^{2} \)
$79$ \( -148576 + 412 T + T^{2} \)
$83$ \( -403408 - 464 T + T^{2} \)
$89$ \( 717912 + 1746 T + T^{2} \)
$97$ \( 140684 - 856 T + T^{2} \)
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